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Homework Help: Integration by Substitution

  1. Nov 30, 2004 #1
    I'm stuck on how to advance further on this problem and if anyone can point my in the right direction I would be greatly appreciative.


    The integral has to be solved using substitution, but we are required to use

    From this:

    But I am stuck on how to convert the remaining portion of the function in terms of du.
  2. jcsd
  3. Nov 30, 2004 #2


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    I gave it a try and couldn't get anywhere with it. Maple says the answer is arcsin(2x-1).

    Is that exactly how the question was given?
  4. Nov 30, 2004 #3
    [tex]u = \sqrt{x}[/tex]


    [tex]u^2 = x[/tex]


    [tex]2du = \frac{dx}{\sqrt{x}}[/tex]

    First use the third equation, then use the second equation to get rid of any other instances of x that're left.

    And Shawn is not correct in his solution.

    Last edited: Nov 30, 2004
  5. Dec 1, 2004 #4
    Shaun's solution looks good to me, what do you propose the actual answer is Justin?
  6. Dec 1, 2004 #5
    Complete the square within the square root in the denominator and the apply the result

    [tex]\int\frac{dx}{\sqrt{a^2-x^2}} = arcsin\frac{x}{a} [/tex]

  7. Dec 1, 2004 #6
    [tex]\int\frac{dx}{\sqrt{x(1-x)}} = 2 \arcsin{\left(\sqrt{x}\right)}[/tex]

    Differentiate it and you'll get the integrand.

    The derivative of arcsin(2x-1) is [tex]\frac{2}{\sqrt{4x^2 - 4x + 2}}[/tex].

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